No. 27 / October Ebert, Michael / Kadane, Joseph B. / Simons, Dirk / Stecher, Jack D. Disclosure and Rollover Risk

Transcription

1 No. 27 / October 2017 Ebert, Michel / Kdne, Joseph B. / Simons, Dirk / Stecher, Jck D. Disclosure nd Rollover Risk

2 Disclosure nd Rollover Risk Michel Ebert 1 Joseph B. Kdne 2 Dirk Simons 3 Jck D. Stecher 4 Jnury University of Pderborn, michel.ebert@uni-pderborn.de 2 Crnegie Mellon University, kdne@stt.cmu.edu 3 University of Mnnheim, simons@bwl.uni-mnnheim.de 4 Crnegie Mellon University, jstecher@cmu.edu

3 Abstrct This pper studies whether nd to wht extent trnsprent disclosure prevents inefficient liquidtion rising from rollover risk. We model n illiquid but solvent borrower who cn design public signl bout wht creditors cn recover from forcing liquidtion, nd wht their clims would be worth if the firm survives. We find tht the signl structure tht minimizes rollover risk never identifies liquidtion or continution vlues, nd tht borrowers cn commit to this structure. Moreover, if creditors cn impose disclosure requirements, they my increse inefficient liquidtion, in order to pool sttes to increse the mount they expect to recover from defults. Keywords: Byes correlted equilibrium, Byesin persusion, disclosure, informtion design, risk dominnce, rollover risk, sender-receiver gmes, strtegic uncertinty, unrveling

4 1 Introduction This pper ddresses the concern tht profitble firm cn go bnkrupt s result of creditors seizing collterl rther thn providing continued funding, out of fer tht too few other climnts will gree to roll their debts over. Our primry interest is in wht informtion n illiquid yet solvent firm cn provide its creditors in order to minimize this rollover risk. Does disclosing informtion bout wht creditors stnd to recover from seizing their collterl provide ressurnce, or does it destbilize? Wht bout disclosures concerning the expected vlue of creditor s clim if the firm survives? If borrower or policymker interested in voiding inefficient liquidtion could design the type of informtion tht creditors would receive in the event of liquidity shock, wht informtion would the idel signl convey to creditors? Secondrily, we re interested in whether creditors would necessrily prefer hving borrowers disclosures minimize rollover risk. In other words, in the bsence of other motives such s contrcting benefits, would creditors necessrily dopt the viewpoint tht no one gins from inefficient liquidtion? We show tht the optiml signl structure for preventing inefficient liquidtion is imprecise, identifying neither the mount creditors cn recover from forcing liquidtion nor the expected vlue of their clims should the firm continue to operte. The intuition is tht n imprecise disclosure pools sttes with low rollover risk with those in which rollover risk is high. In fct, this reltionship turns out to be quite strong: s fundmentl risk increses in sense we mke precise, the optiml signl structure (from the viewpoint of minimizing inefficient liquidtion) becomes less informtive, nd the probbility of inefficient liquidtion flls. Although this line of rgument is fmilir from the globl gmes literture (Crlsson nd vn Dmme 1993, Morris nd Shin 2002, 2004, mong others), it is nturl to expect the borrower s disclosure problem to unrvel, tht is, for the borrower to disclose everything due to creditor skepticism over omissions. We find tht this is not the cse, nd insted find tht 1

5 borrowers cn credibly choose disclosure policies tht minimize rollover risk. If borrower cn completely eliminte rollover risk through (non)disclosure policy, then this policy is lso optiml for the firm s creditors. However, if some chnce of inefficient liquidtion remins fter ny disclosure policy, then creditors interests diverge from borrowers interests, with creditors preferring disclosures tht bring bout some voidble inefficient liquidtion. Creditors cre bout the circumstnces under which they fce rollover risk, nd trde off the probbility of inefficient liquidtion ginst the mount they stnd to lose. The messge is tht we cnnot tke for grnted tht creditor-bcked disclosure requirements generte stbility, or tht disclosures tht reduce inefficient liquidtion increse socil welfre. We develop these results in disclosure nd rollover risk gme between representtive borrower ( she ) nd her creditors ( generic one lbeled he ). The borrower, hving suffered liquidity shock prior to the strt of ply, depends on the creditors willingness to roll their obligtions over. Ech creditor independently decides whether to roll his debts over or to refuse nd seize ny collterl he cn, forcing the borrower into bnkruptcy. As in the rollover risk model of Morris nd Shin (2004), creditor who rolls the debt over when nother creditor forces the borrower into liquidtion receives prt of the bnkruptcy settlement, which is less thn wht the creditor would hve received from seizing the collterl before the bnkruptcy process begins. If no one forces inefficient liquidtion, the borrower nd the creditors ll benefit. In sum, the skeleton of our gme, prior to incorporting symmetric informtion nd disclosure, is the stg hunt gme of Rousseu (discussed in vn Huyck et l. 1990, Crwford 1991, vn Huyck et l. 1993, Bttlio et l. 2001, Skyrms 2001). Our gme builds on this stg hunt structure by introducing n erlier stge, in which the borrower cn costlessly obtin privte informtion bout two distinct but relted vlues. The borrower observes her creditors expected pyoff from rolling their debts over in the event of no forced liquidtion. We cll this the creditors continution vlue. In ddition, the borrower lerns the mrket vlue of the collterl tht creditors cn seize, which we cll the creditors liquidtion vlue. Intuitively, borrower who fces liquidity shock would 2

6 try to discover how strong her creditors tempttion to liquidte is, nd how esily she cn entice her creditors to llow her to py them lter. The borrower issues public disclosure bout these vlues, which my be vgue or incomplete but must be non-frudulent. The creditors then mke their rollover decisions, using risk dominnce s n equilibrium selection criterion (Hrsnyi nd Selten 1988). We focus on risk dominnce becuse it is grounded in the literture on evolutionry gme theory (Crwford 1991, Ellison 1993, Kndori et l. 1993, Young 1993, Temzelides 1997) nd on globl gmes (Crlsson nd vn Dmme 1993, Morris nd Shin 2004 in rollover risk context), nd hs support from lbortory experiments (Schmidt et l. 2003, Anctil et l. 2004, 2010). Risk dominnce cptures the ide of creditors minimizing the negtive effects of being wrong bout equilibrium selection. Our borrower s problem is one of informtion design, recent literture which Tnev (2016) develops fully s multiple receiver extension of the Byesin persusion literture tht Kmenic nd Gentzkow (2011) pioneer. Bergemnn nd Morris (2016b) provide good overview, nd give detils on the importnt building blocks in Bergemnn nd Morris (2016). Informtion design, like mechnism design, hs two components. A bsic gme consists of set of plyers, their pyoff functions, nd their vilble ctions, long with common prior over set of pyoff sttes. An informtion structure cn be thought of s distribution of signl bout the stte. 1 In mechnism design, the designer chooses the bsic gme, tking the informtion structure s given. Informtion design is dul to mechnism design, in the sense tht the designer chooses the informtion structure nd tkes the bsic 1 We limit ttention to the cse in which the borrower designs public signl, nd in which the creditors do not hve dditionl privte informtion. As Wng (2015), Bergemnn nd Morris (2016,b) point out, n informtion designer prefers public communiction when trying to increse coordintion nd prefers privte communiction when the receivers ctions re strtegic substitutes. For exmples of the ltter cse, see Novshek nd Sonnenschein (1982), Vives (1984), Gl-Or (1985), who show the limittions on decentrlized informtion shring in oligopoly, nd the more recent informtion design pproch in Bergemnn nd Morris (2013) nd Micheli (2017), in which n informed designer cn increse ggregte informtion with privte signls. A thorough discussion of privtely informed receivers is beyond our scope, nd we refer the interested reder to Kolotilin et l. (2015). 3

7 gme s given. Our finding tht the solution to the borrower s informtion design problem is to provide n imprecise report goes ginst stndrd unrveling rguments (Grossmn nd Hrt 1980, Milgrom 1981, Grossmn 1981). The borrower is commonly known to be informed, nd therefore cnnot void disclosure by pleding ignornce, s in the models of Dye (1985), Jung nd Kwon (1988), Ben-Porth et l. (2014). Disclosure for her is costless, so she cnnot justify omissions by ppeling to direct or indirect disclosure costs, s in the setting Verrecchi (1983) studies. We usully expect fully informtive report from sender, given n objective of mximizing firm vlue, s Beyer et l. (2010) discuss in their survey rticle. Hedlund (2017) shows tht form of this rgument pplies in Byesin persusion models. A borrower s pyoff, however, is not strictly monotone in how encourging her news is. Providing slightly better news does not benefit the borrower unless doing so cn lter her creditors liquidtion decisions. The flt regions in borrower s pyoff mke imprecise disclosure credible, similr to n observtion tht Milgrom (2008) mkes. The creditors thus do not need to interpret omissions skepticlly (s in Milgrom nd Roberts 1986, Okuno-Fujiwr et l. 1990). Anlogous to the rgument Chen et l. (2008) mke in chep-tlk setting bsed on Crwford nd Sobel (1982), we find tht lck of incentive to seprte limits the mount of voluntry disclosure. Our finding tht creditors my prefer disclosures tht do not minimize rollover risk is to the best of our knowledge new. As we note bove, this result rises if the borrower s disclosure cnnot completely eliminte rollover risk; otherwise, the gme is one of pure common interest. Relted results re in Bouvrd et l. (2015), who study systemic risk. In their setting, if some rollover risk is inevitble, the sender my relese informtion tht delibertely forces some borrowers into inefficient liquidtion, in order to void system-wide collpse. Like us, Bouvrd et l. therefore show tht common interest between the designer nd (some of) the other plyers my vnish if only some coordintion filure is voidble. 4

8 Even if our borrower cnnot eliminte rollover risk, she lwys opts for imprecise disclosure. This result my seem to conflict with Lehrer et l. (2010), who rgue tht incresed informtion precision is welfre improving in common interest gmes. The reson for this pprent difference is tht their notion of welfre is bsed on the best equilibrium ttinble under given informtion structure. Our results, on the other hnd, re bsed on n equilibrium selection criterion: rther thn focusing on the best possible equilibrium, we concentrte on the highest Preto-rnked risk dominnt equilibrium. The reminder of this pper is s follows. Section 2 presents the model. Section 3 gives the results on the optiml choice of informtion structure. Section 4 gives results on the effects of fundmentl risk on the optiml informtion structure. Section 5 concludes. Proofs re in n ppendix. 2 The Model We show our results in model, which consists of bsic gme G nd n informtion structure S, following the nottion of Bergemnn nd Morris (2016). The bsic gme G includes the economic environment (plyers, pyoff sttes, nd common prior over the pyoff sttes) s well s chrcteriztion of the strtegic interction (utilities for the plyers nd their vilble ctions). The informtion structure S consists of public signl t tht depends on the pyoff stte nd tkes vlues in some set T. There re three gents in the model, borrower (she) nd two creditors (both he). The creditors re plyers in the bsic gme G, nd re both risk neutrl nd hve rtionl expecttions. The borrower is designer, who chooses the informtion structure S. Her pyoff depends on the equilibrium tht the plyers in the bsic gme select. We might think of the borrower s plyer 0, but in terms of G the set of plyers includes only the two creditors. We denote the plyers in bsic gme G s I = {1, 2}. 5

9 Our interest is in rollover risk, rther thn the cost of cpitl, credit rtioning, or cpitl structure choices. We therefore ssume tht the creditors hve lons in plce to the borrower t the beginning of ply. As in the rollover risk model of Morris nd Shin (2004), we ssume tht the lons re bcked by collterl. Unlike them, but like the study of sset impirment reporting by Göx nd Wgenhofer (2009), we tret the vlue of the collterl l s stochstic. For simplicity, we ssume the collterl for both creditors is of n identicl nture nd perfectly correlted. Thus, single vlue l represents the vlue of ech creditor s collterl. We refer to this vlue s the liquidtion vlue. If the creditors do not seize their collterl nd insted roll their debts over, the borrower cn remin in business. In this cse, ech creditor owns clim with expected net present vlue c, which we refer to s the continution vlue. This continution vlue is lso stochstic, s is common in bnk run models bsed on Brynt (1980) nd Dimond nd Dybvig (1983) (exmples include Chri nd Jgnnthn 1988, Villmil 1991, Alonso 1996, Hzlett 1997, Kpln 2006, Böckem nd Schiller 2017). 2 The relized pir (l, c) is the pyoff stte. We ssume tht ( l, c) hs commonly known prior ψ. For some, b R, we let the support of ψ be [, b] [, b]. The pyoff stte is known to the designer, our borrower, nd ffects the utilities of the creditors when they ply the bsic gme the G. Prior to the reliztion of (l, c), the borrower chooses the set of signls tht the creditors cn observe, which we cll T, nd function t tht mps ech relized pyoff stte (l, c) to n outcome in T. This is thought of s the borrower s disclosure policy. If t is constnt function, then the borrower discloses nothing bout the pyoff stte. If #(T ) is t lest s lrge s the set of fesible vlues of ( l, c) nd t is injective, then the borrower s policy is full disclosure bout the pyoff stte. If for some positive integer n 2, T = {T 1,..., T n } 2 But not lwys. Adão nd Temzelides (1998) study setting with riskless continution vlue nd no ggregte risk t n interim stge. They obtin non-sunspot coordintion filures becuse mixed-strtegy equilibrium survives forwrd induction refinement, wheres no-run equilibrium does not. 6

10 nd for ll i {1,..., n}, the prior probbility of {(l, c) t(l, c) = T i } is less thn 1, then the borrower s policy is nondegenerte finite prtition of the set of pyoff sttes. We let S = (t, T ) denote the borrower s chosen informtion structure. The creditors lern the reliztion of t nd updte their prior on the stte before plying the bsic gme G. We let t denote the rndom vrible t prior to the reliztion of ( l, c). Two remrks re in order. First, in study of illiquidity, it is nturl to impose tht l c, tht is, tht the borrower is never insolvent. For most of the nlysis below, we mke this ssumption, restricting the support of ψ to {(l, c) [, b] [, b] l c}. An lterntive pproch would llow for the possibility of insolvency ex nte, sy by mndting full disclosure whenever c < l, i.e., requiring n insolvent borrower to declre bnkruptcy. Absent bnkruptcy declrtion, the creditors know tht the borrower is solvent, though rollover risk could remin. This pproch introduces extr complexity, becuse even if l nd c re ex nte independent nd identiclly distributed (iid), t n interim stge they would no longer be so. However, this extr complexity mkes it simple for us to describe n increse in fundmentl risk in the sense of second-order stochstic dominnce, enbling us to ddress whether n increse in fundmentl risk mkes trnsprency more or less desirble. When we turn to this issue, we tke this lterntive pproch. Second, in most of wht follows, we do not require the set of fesible vlues of ( l, c) to be unbounded or to llow for the possibility of unlimited libility. In order to understnd the robustness of our results, however, we find it useful to provide conditions on ψ under which fully disclosing either l or c (but s we shll see, not both) cuses no hrm. Additionlly, in discussing the effects of increses in risk, llowing the set of outcomes to be unbounded enbles us to discuss the cse where ( l, c) re jointly normlly distributed. This specil cse enbles us to mke some observtions relted to first-order stochstic dominnce. Other thn when ddressing these issues, we fix ttention on the cse in which 0 < < b <, s would be expected for the problem we study. 7

11 To complete the description of the bsic gme G, we need to specify the ctions vilble to the plyers nd the pyoffs ech plyer receives from given ction profile nd relized pyoff stte. Ech plyer s set of possible ctions consists of two pure strtegies: A = {R, W }, with strtegy R interpreted s rolling the debt over nd W s withdrwing nd seizing the collterl l. In stte (l, c), the pyoff to plyer i I of strtegy profile ( i, i ) A 2 is l, if i = W u i ( i, i, l, c) = 0, if ( i, i ) = (R, W ) (1) c, if ( i, i ) = (R, R) We view the pyoff to creditor who rolls his debt over when the other creditor forces liquidtion s the mount received in the bnkruptcy settlement. In (1), we normlize this mount to 0. One could justify insted setting this vlue to some mount ε [0, l), so tht the tempttion to seize collterl rther thn risk receiving bnkruptcy settlement is not s strong, though still present. For our purposes, this chnge would not mtter. However, resercher concerned with the rte of convergence to the risk dominnt equilibrium might be interested in how strong bsin of ttrction it is. Binmore et l. (1995) nd Ellison (2000) ddress this issue, nd we refer the interested reder there. It is lso noteworthy tht creditor who forces liquidtion receives the sme mount, given the stte, regrdless of whether the other creditor seizes his collterl or rolls his obligtion over. This implies tht creditor who seizes his collterl does not receive ny dditionl pyment from the bnkruptcy settlement; the pledged ssets essentilly resolve his clims. This is purely convenience; we need not go quite this fr. It is enough to ssume tht, for some δ [0, l], creditor who seizes his collterl lso receives δ from the bnkruptcy settlement. Tht is, we cn modify (1) to mke u i (W, R, l, c) = l + δ. Any vlue of δ within this region would not fundmentlly lter our nlysis. 3 3 To see this, fix reliztion l of the liquidtion vlue, let δ [0, l], nd for ech i I, let the pyoff u i (W, R, l, c) = l + δ. Suppose c is the relized continution vlue. As we discuss below, strtegy profile (R, R) is risk dominnt under full disclosure if nd only if c/l exceeds threshold. It is strightforwrd to 8

12 The borrower s objective is to keep her job. Her pyoff depends on the ction profile ( 1, 2 ) A 2 tht the creditors choose in bsic gme G s follows: 1, if 1 = 2 = R u 0 ( 1, 2 ) = 0, otherwise Her pyoff is thus function of the equilibrium the creditors ply, nd depends only indirectly on the stte through its effect on the equilibrium in the bsic gme G. To try to ffect equilibrium behvior, the borrower chooses the informtion structure S = {T, t}, s described bove. If the set of fesible pyoff sttes includes reliztions with l > c, then we include in T trivil disclosure corresponding to bnkruptcy declrtion in the event l > c, in which cse the creditors would not ply bsic gme G. Otherwise, given (G, S), the creditors observe the relized signl t, updte their beliefs bout the pyoff stte ( l, c) using t nd their common prior ψ, nd choose their strtegies. It is commonly known tht the creditors choose their strtegies bsed on risk dominnce, given their posterior beliefs bout the stte (using pyoff dominnce to brek ties). Tht is, there is decision rule σ tht recommends strtegies to the creditors, which depends only on the signl: 1, if (E[(u i (W, R, l, c) t] E[u i (R, R, l, c) t]) 2 σ((r, R) t) = (E[(u i (R, W, l, c) t] E[u i (W, W, l, c) t]) 2 0, otherwise (2) (3) nd σ((w, W ) t) = 1 σ((r, R) t) In words, the decision rule is to ply equilibrium (R, R) if nd only if the product of the creditors losses of unilterl devition from (R, R) wekly exceeds the product of the show tht, s δ increses to l, this threshold increses to 3, nd s δ decreses to 0, this threshold flls to 2. By setting δ = 0, we stck the deck ginst our finding tht full disclosure is efficient, s this ssumption mkes the probbility tht full disclosure leds to selection of the pyoff dominnt equilibrium s lrge s possible. 9

13 devition losses from (W, W ). After observing signl t, the creditors updte their beliefs bout the stte nd bse their decisions on their expected utilities. Substituting (1) nd rerrnging, we obtin tht σ((r, R) t) = 1 if nd only if E[ c t] 2E[ l t] (4) Given tht the borrower cnnot control the decision rule σ, her informtion design problem is therefore to choose the informtion structure S in order to mximize the prior probbility ψ tht (4) holds. Note tht the signl t nd the decision rule σ correlte the strtegies of the creditors, nd tht ech creditor finds it optiml to obey decision rule σ given the reliztion of t. Bergemnn nd Morris (2013, 2016,b) dub this equilibrium concept Byes correlted equilibrium, nd the willingness to follow σ n obedience condition, nlogous to n incentive comptibility condition in mechnism design. In our gme, σ perfectly correltes the creditors ctions: they lwys choose the risk-dominnt equilibrium, nd they lwys hve the sme (firstnd higher-order) beliefs bout which equilibrium is risk dominnt. Thus, while σ does not lwys choose the Preto-dominnt equilibrium, it never leds to mis-coordintion (i.e., mismtched strtegies). 3 Results 3.1 Benchmrk: Full disclosure We begin by considering the benchmrk of full disclosure, tht is, of setting the signl t = ( l, c). For now, we will ssume tht the problem is one of pure liquidity risk, i.e., tht the set of fesible outcomes is {(l, c) [, b] [, b] l c}. We will lso restrict ttention to the cse with 0 < b <, 10

14 It is immedite tht in ny nondegenerte setting, full disclosure cnnot ttin first-best: there is lwys some inefficient liquidtion if the the borrower revels ll her informtion. Figure 1 illustrtes. c b c = 2l c = l b l Figure 1: Continution vlue c versus liquidtion vlue l. Above the 45 line, the borrower is solvent nd continution is efficient (shown in gry). It is risk dominnt to roll the debts over if nd only if (l, c) is bove the c = 2l line (dotted gry re). With full disclosure, inefficient liquidtion occurs whenever (l, c) lies in the region between the c = l nd the c = 2l lines (gry qudrilterl). In the figure, the right tringle with vertices {(, ), (, b), (b, b)}, shown in gry, represents the entire region in which the borrower is solvent nd rolling the debt over is Pretodominnt. From (4), we see σ((r, R) l, c) = 1 if nd only if c 2l. This is shown in the dotted tringle, bounded by the c = 2l line, with vertices {(, 2), (, b), (b/2, b)}. The gry (undotted) irregulr qudrilterl with vertices {(, ), (, 2), (b/2, b), (b, b)} is the region in which σ selects the inefficient liquidtion equilibrium (W, W ). Unless ψ ssigns zero 11

15 probbility to this region, full disclosure informtion must hve efficiency loss. It lso immedite tht there re informtion structures tht chieve coordintion on (R, R) with higher probbility thn full disclosure. To the extent tht the borrower cn void n unrveling problem, she cn genericlly do better thn choosing full disclosure. Refer gin to Figure 1. The centroid of the full disclosure region hs coordintes (E[ l c 2 l], E[ c c 2 l]) Except in the degenerte cse in which ψ ssigns zero probbility to every point strictly bove the c = 2l line, this centroid is in the interior of the full disclosure region. The borrower cn then consider n informtion structure tht pools the entire full disclosure region with sufficiently smll region djcent to nd just below the c = 2l line. Cll the full disclosure region Φ nd the dditionl smll region djcent to it, choosing so tht the centroid of Φ is still on or bove the c = 2l line nd so tht ψ ssigns nonzero probbility to. Define new rndom vrible τ by 1, if (l, c) Φ τ(l, c) = 0, otherwise Suppose the borrower chooses signl τ insted of t, nd nnounces t the strt of ply tht she will use this informtion structure. After observing the relized signl τ, the creditors choose their strtegies ccording to risk dominnce. Then (3) becomes σ((r, R) τ = 1) = 1 if E[ c τ = 1] > 2E[ l τ = 1], which holds by construction. Similrly, becuse outside of Φ it is lwys the cse tht c < 2l, we hve σ(w, W τ = 0) = 1. Therefore, decision rule σ selects equilibrium (R, R) with the probbility ψ ssigns to Φ, which is strictly greter thn the probbility ssigned to Φ, i.e., of selecting the pyoff-dominnt equilibrium under full disclosure. 12

16 3.2 Optiml informtion structure for minimizing rollover risk The nlysis bove shows tht full disclosure cnnot be the borrower s optiml decision choice unless the borrower s problem unrvels. We now turn to the problem of finding disclosure tht mximizes the probbility tht both creditors roll their debts over, nd then verifying tht this informtion structure is one tht does not unrvel. To limit the scope of the borrower s problem, we first observe tht σ recommends decisions in pure strtegies for ech plyer, tht these strtegies re perfectly correlted, nd tht the number of pure strtegies vilble to ech plyer is 2. Therefore, it suffices to restrict ttention to informtion structures with T = 0, 1, so tht t prtitions the set of pyoff sttes into two subsets. Further, we will tret prtitions s equivlent if they re equl on ll but set of mesure zero, s the creditors mximize their expected pyoffs given their informtion. The following exmple illustrtes the ide. As we discuss bove, n informtion structure tht provides full disclosure is one in which the relized signl t is sufficient sttistic for (l, c). It is possible to replicte wht the borrower chieves under full disclosure, in every pyoff stte, with the following informtion structure: 1, if c 2l t(l, c) = 0, if c < 2l Substituting the relized signl t defined in (5) into (4), we immeditely obtin tht σ((r, R) t) = 1 if nd only if c 2l, exctly s in the full disclosure cse. We now show tht the optiml informtion structure for the borrower in generl does not identify the liquidtion vlue l or the continution vlue c. (5) Theorem 1 The optiml informtion structure from the borrower s viewpoint, tht is, the informtion structure tht minimizes rollover risk, is either nondisclosure or threshold disclosure. In the ltter cse, the threshold depends on the reltionship between c nd l but not the individul reliztions of c or l. 13

17 Figure 2 illustrtes the ide of Theorem 1 in the cse in which ψ is uniform distribution over the tringle M with vertices {(, ), (, b), (b, b)} (the gry region in the figure). By pooling sttes below the c = 2l line with those bove it, the borrower increses the probbility of continution. She cn increse this pooled region R until its centroid meets the c = 2l line. Beyond tht point, pooling R with ny dditionl subset of M with positive probbility mesure moves the centroid of R below the c = 2l line, cusing inefficient liquidtion. b c c = 2l R c = 2l + b/2 c = l b l Figure 2: Specil cse in which ( l, c) is uniformly distributed over the upper gry tringle. In the trpezoid between the c = 2l nd the c = 2l + b/2 lines (drk gry), full disclosure leds to inefficient liquidtion. However, restricting disclosure to whether c 2l + b/2 prevents inefficient liquidtion in the drk gry nd dotted regions. The boundry of region R in Figure 2, s shown in the proof of Theorem 1, is the line c = 2l + b/2. Observe tht, if ψ is uniform over the tringle M, then this shifted boundry is prllel to the c = 2l line tht bounds the full disclosure region. A strtegy of fully disclosing l nd not c, by contrst, would pper in this picture s verticl line. Anlogously, 14

18 disclosure of c nd not l would pper s horizontl line. For ny distribution tht is bsolutely continuous with respect to Lebesgue mesure, disclosure of l or of c nd not the other vlue would therefore be report of set tht is thin, i.e., of mesure zero. Thus, disclosure of single component of (l, c) provides little vlue to the borrower. Either nondisclosure is lso optiml, so tht distinguishing set of mesure zero is inconsequentil, or nondisclosure is suboptiml, in which cse distinguishing set of mesure zero does not help improve the probbility of inefficient continution. 3.3 Optiml informtion structure, creditors viewpoint We now consider the informtion structure tht the creditors would prefer, if they could impose hving someone cting on their behlf s the informtion designer. Although the creditors would lso like to prevent inefficient liquidtion, their objective is different. They cre bout how much they recover, nd this gives them preference for inefficient liquidtion in some sttes over others. Letting P r( ) be the probbility mesure ssocited with ψ, we cn write the creditors idel informtion design s the solution of the following problem subject to the obedience condition mx E[ c ˆR]P r( ˆR) + E[ l M\ ˆR](1 P r( ˆR)) (6) ˆR M E[ c ˆR] 2E[ l ˆR] i.e., the centroid of ˆR must lie on or bove the c = 2l line. If M\ ˆR hs positive probbility, then this constrint binds (the creditors do not benefit from wste). We now show tht this difference in objective mens tht the creditors would prefer different disclosure region, unless rollover risk is completely voidble (in which cse there is lwys nondisclosure) or completely unvoidble (in which cse disclosure is irrelevnt). We illustrte this in the cse in which ψ is uniform over the efficient continution region M. To 15

19 rule out the cses in which ll prties prefer nondisclosure nd those in which disclosure is irrelevnt, we ssume 0 < 2 < b. Purely s technicl convenience, we lso ssume b 4 (lthough the rgument goes through provided b <. We then hve the following: Theorem 2 Under the ssumption tht ψ is uniform over M nd 0 < 2 < b < 4, the optiml informtion structure from the creditors viewpoint differs from the optiml informtion structure from the borrower s viewpoint. Figure 3 illustrtes the conflict between the borrowers nd creditors. The optiml region ˆR from the creditors viewpoint is the right tringle with vertices {(, ŷ), (, b), (ˆx, b)}. We see from the figure tht, compred with the borrower, the creditors re willing to tolerte more inefficient liquidtion when both l nd c re low, if in exchnge they cn void some dditionl inefficient liquidtion when c is high nd l is not too high. Lstly, we note the following: Proposition 1 Suppose tht either t = 1 if nd only if the pyoff stte (l, c) is in the borrower s optiml disclosure region, or t = 1 if nd only if the pyoff stte (l, c) is in the creditors optiml disclosure region. Then there is no unrveling: fter the creditors observe ny reliztion of t, the borrower hs no incentive to revel ny dditionl informtion. To summrize, we find tht full disclosure is never optiml. Nondisclosure, on the other hnd, my be optiml, nd if it is, it is optiml from both the borrower s nd the creditors viewpoint. If nondisclosure is not optiml, then the best informtion structure from the borrower s viewpoint, tht which minimizes rollover risk, generlly is suboptiml from the creditors viewpoint. This is becuse creditors re concerned with the expected costs of rollover risk, wheres borrowers re concerned with the overll mount of rollover risk, i.e., the probbility of coordintion filure. 16

20 c R b ŷ ˆx b l Figure 3: The creditors prefers to hve the borrower disclose whether (l, c) is inside the region bounded by the tringle with vertices {(, ŷ), (, b), (ˆx, b)}, with the dshed line s the boundry. This region is smller thn the borrower s preferred disclosure of whether (l, c) R. This greter probbility of wste is offset for the creditors by higher expected pyoffs in liquidtion nd in continution. 4 Effects of risk on (non)disclosure The objectives of borrowers nd creditors re ligned whenever creditors would roll their debts over if the borrower discloses nothing. In this section, we investigte the effects of the fundmentl risk ssocited with the pyoff stte s distribution ψ on whether nondisclosure is optiml. We focus on nondisclosure in this setting becuse it chieves first-best, voiding ny concerns over whether optimlity is tken from the borrower s or creditors viewpoint. The notion of fundmentl risk is nunced with multiple dimensions, s is well-known (see, 17

21 e.g. Kihlstrom nd Mirmn 1974, 1981). To void the subtleties tht rise in discussing the risk ssocited with joint distribution ψ, we mke smll but crucil modifiction to the economic environment. Specificlly, we now ssume tht ex nte, both l nd c re independently nd identiclly distributed ccording to some unidimensionl cumultive distribution function F with support [, b]. This llows for the possibility, gin ex nte, tht the borrower could become insolvent. Our focus remins entirely on inefficient liquidtion, which cn occur only if the reliztion of c is t lest s lrge s tht of l. Otherwise, liquidtion is efficient, the firm declres bnkruptcy, nd the gme ends. We cn now ssocite risk with the ex nte distribution F. Given tht the bsic gme G is reched, the creditors updte their beliefs bout the distributions of l nd of c, even before the borrower revels ny informtion. We write the cumultive distribution functions (cdfs) nd probbility density functions (pdfs) of l nd c respectively s F L (x) = 1 [1 F (x)] 2 f L (x) = 2[1 F (x)]f(x) (7) F C (x) = F 2 (x) f C (x) = 2F (x)f(x) (8) As noted in Section 2, this dditionl ssumption still mens tht, bsent bnkruptcy declrtion, the creditors know tht the vlue of c must be t lest s high s tht of l. Nevertheless, this dded ssumption is not innocuous, becuse now l is the lower of two drws (the first smple order sttistic), nd c is the higher (the second smple order sttistic). The creditors cnnot tret the two s independent, nd the borrower knows this when designing the informtion structure. In wht follows, we refer to the ex nte distribution F s the implicit risk distribution. Figure 4 illustrtes the informtion tht c provides bout l nd tht l provides bout c. The fct tht the liquidtion vlue nd continution vlue re, in the current setting, informtive bout ech other requires us to revisit the implictions of disclosing either vlue while 18

22 Figure 4: The distributions of the continution nd liquidtion vlues. l c F c c x 1 = l x 2 = c b l c l F b l b leving the other undisclosed. By disclosing c, the borrower implicitly is now lso telling the creditors tht l c, nd this dditionl informtion content could chnge our ssessment bout whether it is ever optiml for the borrower to mke single-coordinte full disclosure. We show below tht this concern turns out not to mtter, i.e., tht it remins the cse tht it is optiml to disclose either c or l (but not both) only if nondisclosure is lso optiml, nd tht the converse is flse in ech cse. Thus, our erlier ssessment tht disclosing c or l is either superfluous or hrmful is robust to this modifiction of our ssumptions. We begin by showing tht nondisclosure cn be optiml. Theorem 3 chrcterizes the optimlity of nondisclosure in terms of the underlying distribution. The result comes from the fct tht, given the bsence of bnkruptcy declrtion, the borrower s continution vlue is the higher of two smple order sttistics. Theorem 3 Let µ be the men of implicit risk distribution F ( ). Then it is optiml for the borrower to disclose nothing if nd only if E[ c] 4 3 µ (9) Theorem 3 suggests tht strtegic risk is vnishes once fundmentl risk is sufficiently high. We now show tht this is the cse: greter risk, in the sense of second-order stochstic dominnce, is ssocited with mking nondisclosure optiml. Lemm 1: 19

23 Lemm 1 Suppose x 1, x 2 re iid, ỹ 1, ỹ 2 re iid with men zero nd with nondegenerte distribution, nd the x i nd ỹ i re mutully independent. Then E[mx{ x 1 + ỹ 1, x 2 + ỹ 2 }] > E[mx{ x 1, x 2 }]. Corollry 1 Suppose F is n implicit risk distribution under which nondisclosure chieves first-best, i.e. for which Inequlity (9) holds. Let H be men-preserving spred of F. Then nondisclosure chieves first-best under H. The converse does not necessrily hold. The intuition of Corollry 1 is tht risk is n rgument for nondisclosure, not for incresed disclosure. A similr result holds with incresing Vlue-t-Risk. We restrict ttention here to normlly distributed pyoffs, s our purpose here is only to provide n illustrtion. In this specil cse, the result follows from technicl proposition Proposition 2 Suppose n underlying risk distribution F is norml distribution with men µ nd vrince σ 2. Then distribution ˆF represents higher Vlue-t-Risk (i.e., the negtive portion of the distribution is more negtive t every quntile under ˆF thn under F ) if nd only if the vrince under ˆF is lso σ 2 nd the men under ˆF is below µ. In this cse, if nondisclosure is optiml in the sense of Theorem 3, then nondisclosure is lso optiml under ˆF. Lstly, we show tht selective full disclosure of either the liquidtion vlue l or the continution vlue c is either superfluous or hrmful. It is superfluous in some, but not ll, cses in which nondisclosure is optiml, nd hrmful otherwise. The conditions under which disclosing l lone re hrmless turn out to be quite strong: the cdf F must hve infinite vrince. A comprison with Theorem 3 helps clrify the intuition: nondisclosure is optiml if risk is high, but does not require nything nerly s strong s infinite vrince. 20

24 Lemm 2 Suppose tht, for ll vlues of l [, b), E[ c l] = 2l. Then F is Preto distribution with scle prmeter R ++ nd shpe prmeter α = 2: ( ) 2 ( x [, )) F (x) = 1 (10) x If E[ c l] 2l for ll l with strict inequlity for some l, then F is more hevily tiled thn the Preto distribution with scle prmeter 2. Together, these imply tht either there re some vlues of l for which E[ c l] < 2l, or F is n infinite vrince distribution. From Lemm 2, we cn now prove tht disclosure of l never improves on nondisclosure. Proposition 3 Suppose E[ c l] 2l for ll l [, b). Then nondisclosure of both c nd l prevents rollover risk. Tht is, if disclosure of l chieves first-best, then so does nondisclosure. The conditions under which disclosure of the continution vlue lone is superfluous rther thn hrmful re not quite s strong. Nevertheless, we show tht they re sufficient but not necessry for nondisclosure to be optiml. The importnt conditions for disclosing c lone to be hrmless re tht F is wekly concve nd tht 0. The intuition for this lst requirement is s follows: suppose the creditors clims re bcked by collterl with vlue tht is bounded wy from zero. Then disclosure of sufficiently low reliztion of c is sufficient sttistic for (W, W ) being risk dominnt. In other words, if the collterl hs vlue bounded wy from 0, then this bound prevents the borrower from pooling informtion when the overll economic conditions re wek. Lemm 3 Suppose tht, for ll vlues of c (, b], E[ l c] = c/2. Then F is uniform distribution with lower bound t 0. If E[ l c] c/2 for ll c with strict inequlity for some c, then F is wekly concve nd bounded, with nonpositive lower bound. Compring Lemm 3 nd Theorem 3, we see tht the optimlity of disclosing c depends on the conditionl expecttion of l, nd the optimlity of nondisclosure depends on the 21

25 unconditionl properties of F. The following result sys tht ny bounded distribution stisfying the conditions of Lemm 3 necessrily stisfies Eqution (9), but tht there re distributions F for which (9) holds but Lemm 3 does not. Proposition 4 Assume >, so tht the worst possible loss is finite. If disclosure of c mkes it risk dominnt to roll the debt over, then so does nondisclosure. The converse is flse. 5 Conclusion Our min findings show tht borrower disclosures concerning rollover risk optimlly do not enble creditors to seprte their potentil gins from keeping the borrower in business from those of forcing the borrower into liquidtion nd recovering the vlue of their collterl (Theorem 1). These optiml disclosures do not require ny externl form of commitment, s they do not unrvel into full disclosure (Proposition 1). If creditors cn impose disclosure requirements, they my do so in wy tht increses the likelihood of inefficient liquidtion in order to ssure tht, when public informtion leds to inefficient liquidtion, the cost is not too high (Theorem 2). However, this conflict of interest occurs only if the credit risk of the borrower is low. Once the fundmentl risk becomes sufficiently high, ll prties prefer tht borrowers disclose nothing beyond the fct tht they still hve going concern vlue (Theorem 3). These results run counter to our intuition bout the conflict between creditors nd equity holders (Jensen nd Meckling 1976, Myers 1977). In rollover risk context, it is the owners of the firm who cre bout defult probbility, nd the creditors who cre bout expected net present vlues. Wht my be more surprising is tht creditors wnt more fundmentl risk when they fce strtegic risk, nd if their borrowers sy nothing bout how these fundmentl risks re 22

26 plying out, then creditors re hppy bout it. To some extent, this is driven by force similr to option vlue: higher fundmentl risk increses the probbility tht there is lrge gp between the potentil gins from rolling debts over nd those from seizing collterl nd cutting one s losses. The other importnt force is tht coordintion is, in principle, similr to collusion. Controlling rollover risk, from n informtion design viewpoint, is problem of constructing informtion sets tht foster collusion, s in n informtion shring problem mong oligopolists. This is inherently different from the informtionl issues we consider in competitive debt mrkets, in which the socil optimum typiclly involves preventing collusion. A Proofs Proof of Theorem 1. If the prior ψ stisfies E[ c] 2E[ l], then the borrower hs nothing to gin from disclosure. In this cse, (4) is replced with prior expecttions, s the signl t is uninformtive bout the pyoff sttes. Rolling the debts over is risk dominnt in this cse, nd chieves first-best. It remins to show tht the optiml prtition does not in generl revel the vlues of c or l. We show this for the cse in which ψ is uniform distribution over the tringle with vertices {(, ), (, b), (b, b)}, s extensions to other cses with tomlesses distributions tht re bsolutely continuous with respect to Lebesgue mesure re strightforwrd. Let M be this tringle, i.e. M = supp ψ. We ssume 0 < 2 < b 4. This is unimportnt nd is only for technicl convenience, except in tht it gurntees tht some inefficient liquidtion is voidble. The borrower s objective is choose the lrgest possible subset R of M with centroid on or bove the c = 2l line. Letting P r be the probbility mesure ssocited with ψ, the 23

27 borrower s objective is ( ) mx P r(r) s. t. E[ c R] 2E[ l R] R M (11) The centroid of M, denoted m := (m l, m c ), is defined s follows: m l = m c = b b b (b 2 x 2 ) b (b x)xdx 2 + b = (b x)dx 3 dx 2 + 2b = (b x)dx 3 The rtio of m c /m l is (2b + )/(2 + b), which is less thn 2 becuse > 0. It is therefore cler tht the borrower would not choose nondisclosure in this cse, s it leds to inefficient liquidtion. Becuse ψ is uniform, strightforwrd rgument (vilble from the uthors on request) shows tht the optiml prtition boundry is liner. For some x, y [, b], consider the line segment tht psses through (, y ) nd (x, b), nd let R be the tringle with vertices {(, y ), (, b), (x, b)}. Our gol in this first prt of the proof is to find the optiml vlues of (x, y ) from the borrower s viewpoint. Anlogous to (12 13), the centroid of R, denoted r = (r l, r c ), is 2 + x r l = 3 r c = y + 2b 3 Becuse the centroid of M is below the c = 2l line, the constrint in (11) binds, i.e., r c = 2r l. Consequently, y + 2b = 4 + 2x (12) (13) (14) (15) y = 4 2b + 2x (16) Under the joint uniformity ssumption, the probbility of R given tht l < c is the rtio of the re of R to the re of M: p = (b y )(x ) (b ) 2 (17) 24

28 Substituting (16) into (17), The first-order condition is p = (b 4 + 2b 2x )(x ) (b ) 2 = ( 2x + 3b 4)(x ) (b ) 2 = 2(x ) 2 + (3b 2)x 3b (b ) 2 (18) 4x + 3b 2 (b ) 2 = 0 (19) The second derivtive is 4, so the solution to (19) gives unique globl mximum. Solving for x, nd therefore x = 3b 2 4 y = 3 b 2 The ssumption tht b 4 gurntees tht y is not below the lower vertex (, 2) of the full disclosure region. Other thn ffecting the shpe of R (mking it right tringle rther thn the union of right tringle nd trpezoid), this ssumption hs no effect, nd the optiml boundry is still prllel to the c = 2l line. The points (, 3 b/2) nd ((3b 2)/4, b) determine the hypotenuse of the region R s the line c = 2l + b/2, s shown in Figure 2. Observe tht this boundry is neither horizontl line (i.e., determined by cutoff vlue of c nor verticl line (i.e., determined by cutoff vlue of l). Proof of Theorem 2. (20) (21) From the creditors viewpoint, the objective is to choose ˆR to mximize the expected pyoff. As shown in (6), the creditors receive the continution vlue if (l, c) ˆR nd the liquidtion vlue otherwise. Define line tht psses through (, ŷ) nd (ˆx, b) for some vlues ˆx nd ŷ. Let ˆR be the tringle with vertices {, ŷ), (, b), (ˆx, b)}. Our gol in this second prt of the proof is to find the optiml vlues of (ˆx, ŷ) from the creditors 25

29 viewpoint. The centroid of ˆR, denoted ˆr = (ˆr l, ˆr c ) is 2 + ˆx ˆr l = 3 ˆr c = ŷ + 2b 3 (22) (23) The expecttion of c given ˆR is ˆr c = (2b + ŷ)/3. Becuse ˆr lies on the c = 2l line, we cn substitute for ŷ nd obtin ˆr c = 2ˆr l = (4 + 2ˆx)/3. Let ˆp be the probbility of ˆR. The centroid ŝ of Ŝ := M\ ˆR is ŝ = so tht the expecttion of l given Ŝ is ( ) ( ) b 2 + ˆx ŝ l = ˆp 1 ˆp 3 3 ( ) 1 (m ˆpˆr) (24) 1 ˆp (25) The creditors objective is therefore to mximize [( ) ( 4 + 2ˆx b E[π] = ˆp + (1 ˆp) ˆp 3 1 ˆp 3 ˆp(2 + ˆx) b = 3 ( 2ˆx + 3b 4)(ˆx )(2 + ˆx) = b 3(b ) 2 3 )] 2 + ˆx 3 (26) The first-order condition is 2ˆx 2 + 2(b 2)ˆx + b (b ) 2 = 0 (27) The negtive root is below zero nd extrneous. The positive root is ˆx = b b + b 2 2 (28) The second order sufficient condition is 4ˆx + 2(b 2) (b ) 2 < 0 ˆx > (b 2) 2 (29) which holds for the positive root but not for the negtive root. 26

30 Compring the creditors preferred vlue ˆx with the borrower s preferred vlue x given in (20), we cn find conditions under which they coincide: b b + b 2 2 = 3b 2 4 2b b + b 2 = 3b b + b 2 = b b + b 2 = 3 2 3b b2 = 0 ( 2) b 2 = 0 (b + 2)2 4 = b2 4 + b + 2 b = 2 (30) By hypothesis, b > 2, contrdicting Eqution (30), which cn hold only if the probbility of efficient continution is zero. Proof of Proposition 1. First observe tht both the borrower s optiml signl structure nd the creditor s optiml signl structure re supersets of the full disclosure region. With either of these structures, the borrower hs nothing to gin from reveling dditionl informtion fter the creditors observe the public signl t. If t indictes tht (l, c) belongs to the region for which σ( ) recommends both creditors roll their debts over, then dditionl disclosures re superfluous if c 2l nd destructive if c < 2l. If t indictes tht (l, c) belongs to region for which σ( ) recommends both creditors force liquidtion, then necessrily c < 2l, nd there is no supplementl informtion tht could lter this decision. Proof of Theorem 3. The pdfs of c nd l re Recll tht nondisclosure is optiml if nd only if E[ c] 2E[ l]. f C (x) = 2F (x)f(x) f L (x) = 2[1 F (x)]f(x) (31) 27

31 This mens 2 6 b b E[ c] 2E[ l] iff 2xF (x)f(x)dx 2 2xF (x)f(x)dx 4 b b 2x[1 F (x)]f(x)dx xf(x)dx 4 b b xf (x)f(x)dx 4µ x E[ c] = 2 b xf (x)f(x)dx xf (x)f(x)dx 4 3 µ x (32) Proof of Lemm 1. Let ( x ( x #, ỹ #, x, ỹ 1, ỹ 1, x 2, ỹ 2 ), x 1 x 2 ) = ( x 2, ỹ 2, x 1, ỹ 1 ), x 2 > x 1 Then x # = mx{ x 1, x 2 }. Then E[ỹ # ] = E[ỹ 1 x 1 x 2 ]P ( x 1 x 2 ) + E[ỹ 2 x 1 < x 2 ]P ( x 1 < x 2 ) Becuse the ỹ i re iid nd independent of the x i, E[ỹ # ] = E[ỹ 1 x 1 x 2 ]P ( x 1 x 2 ) + E[ỹ 1 x 1 < x 2 ]P ( x 1 < x 2 ) = E[ỹ 1 ] = 0, E[mx{ x 1 + ỹ 1, x 2 + ỹ 2 }] = E[mx{ x # + ỹ #, x + ỹ }] = E[mx{ x # + ỹ #, x + ỹ + ỹ # ỹ # }] = E[mx{ x #, x + ỹ ỹ # }] + E[ỹ # ] = E[mx{ x #, x + ỹ ỹ # }] E[ x # ] = E[mx{ x 1, x 2 }] (33) 28

32 Furthermore, (33) holds strictly provided P ( x + ỹ ỹ # > x # ) > 0 Now x + ỹ ỹ # > x # ỹ ỹ # > x # x 0 (34) Suppose first tht the distribution of X hs some tom x in its support, i.e., so tht P (X = x) = p > 0 Then P ( x # = x, x = x) = p 2 > 0 Since the distribution of the ỹ i is nondegenerte, there is positive probbility tht ỹ > 0 nd ỹ # < 0. In this cse, (34) holds with positive probbility, so tht the inequlity (33) is strict. Next, suppose tht the distribution of the x i is purely continuous. Then the proof of the strictness of inequlity (33) hs two steps: 1. For every ε > 0, there is positive probbility tht ε > x # x ε > 0 such tht there is positive probbility tht ỹ # ỹ > ε. To prove the first step, let ε > 0 be given. Let [m, M] be n intervl with m < M with positive X-probbility. For let k { 0,..., 2(M m) ε d k = m + εk 2 29 },

33 The intervls (d k, d k+1 ) re disjoint, nd t lest one hs positive probbility. Then there is positive probbility tht x #, x re in the sme intervl (d k, d k+1 ), nd hence tht ε > x # x > 0 To show the second step, suppose to the contrry tht ε > 0, P (ỹ ỹ # > ε) = 0. Then P (ỹ ỹ # = 0) = 1, violting nondegenercy. Therefore, if X is purely continuous, then with positive probbility, (34) holds. Consequently, (33) holds with strict inequlity. Proof of Corollry 1. Immedite corollry of Lemm 1. Proof of Proposition 2. We first observe tht, if the Vlue-t-Risk is higher under ˆF t every quntile if nd only if F first-order stochsticlly domintes ˆF. Tht is, given normlity, shift of the left til of the distribution implies shift of the entire distribution. Next, we remrk tht, for normlly distributed rndom vrible with men µ nd vrince σ 2, the expecttion of the higher of two independent drws is µ+σ/ π, nd the expecttion of the lower of two independent drws is µ σ/ π. This is known result from the theory of order sttistics. It follows from (4) tht rolling over is risk-dominnt equilibrium if nd only if µ 3σ/ π. It is therefore enough to show tht if norml distribution becomes worse in the sense of first-order stochstic dominnce, the men decreses nd the vrince remins unchnged. In order to demonstrte this, ssume tht F nd ˆF hve mens µ nd ˆµ, with ˆµ < µ. At α = 1/2, we know tht the cdf F is to the right of the cdf ˆF. It remins to show tht if F nd ˆF do not cross, then the vrince under F, denoted σ 2, must equl the vrince under ˆF, denoted ˆσ 2. If, to the contrry, the two cdfs cross t some 30